What is the difference between an Active animated rigid body and a Passive rigid body?
Active objects are the core part of the physics simulation, with dynamic movement, gravity, etc. By checking Active, you are essentially telling the Physics simulator that those objects should experience forces, and it has to figure out where they go over the course of the simulation.

Passive objects, on the other hand, are fixed in place--are static. For the purposes of the simulation, they essentially act as rigid-bodied, static, collision objects. The Animated option allows you to control its motion with keyframes, but it still does not experience the forces modeled on Active objects.

For example, if you're making a simulation of a ball falling onto a table, the ball would be an Active object, since it is affected by gravity, whereas the table would be Passive, since it is fixed in place.